boundary conditions and maxwell equations
DESCRIPTION
ElectromagneticsTRANSCRIPT
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MAXWELLS
EQUATIONSAND BOUNDAR
CONDITIONS
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INTRODUCTION
Maxwells equations characterize macroscopic
matter by means of its permittivity , permeability
, and conductivity , where these properties are
usually represented by scalars and can vary amongmedia.
If the field exists in a region consisting of two
different media, the conditions that the field must
satisfy at the interface separating the media are
called boundary conditions.
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E Electric fieldH Magnetic field
D Electric displacement
B Magnetic induction
r Electric charge density
J Electric current density
e Permittivity
m Permeability
e 0 Permittivity of vacuum
m0 Permeability of vacuum
P Electric polarization
M Magnetization
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Boundary Conditions
for agnetic Field
The derivation of boundary conditions for the
magnetic field, follows similar arguments to
that of the electric field, but using equations:
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Again we consider the normal and
tangential components as illustrated
below.
Normal and tangential components the B field on eitherside of an interface.
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Normal Component of B
The boundary condition for the normal
component of the magnetic field can be
obtained by applying Gausss flux law:
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Now if two planes are allowed to approach one
another, keeping the interface between them, the
area of the curved surface will approach zero,
giving:
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Tangential Component of H
We can derive the tangential component of H
by applying Amperes law
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The variation in H across an interface is
obtained by the application of this law around a
closed rectangular path, as shown in the figure.Assuming no current at the interface, letting the
rectangle shrink to zero:
Thus, tangential component of H has the same
projection along the two sides of the rectangle.
Since the rectangle can be rotated 90 and theargument is repeated, it follows.
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The tangential component of H is continuous across a
current-free interface. The relation
between the angles made byH1
and H2
with a current-
free interface (see figure below) is obtained.
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Current Sheet at
the Boundary
The magnetic field H produced by
this current sheet is given by:
The resultant magnetic field,
This is expressed by the vector
formula
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For reference purposes, the relationship for E and
D across the interface of two dielectrics are shown
below along the relationships for H and B.
agnetic Fields
B
n1= B
n2
H
t1= H
t2(Current-free)
(H1 H2) an12 = K (with current sheet)
(current-free)
Summary of
Boundary Conditions
Electric Fields
D
n1= D
n2(Charge-free)
(D1 D2) an12 = -r (with surface charge)
t1=
t2
(charge-free)
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axwell's equationsare a series of four partial
differential equations that describe the force of
electromagnetism. Individually, the four equations are
named Gauss' law, Gauss' law for magnetism, Faraday's
law and Ampere's law.
General form of axwells equations:
1. It actually applies to all cases, either in vacuum or in amedium. It is thus called the general form of axwells
equations.
2. It is also called axwells equations in vacuum.
AXWELLS
EQUATIONS
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Maxwells Equations, General Set
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Maxwells Equations, Free-space Set
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Thanks!Any questions?
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